Abstract
We rigorously prove the well-posedness of the formal sensitivity equations with respect to the viscosity corresponding to the 2D incompressible Navier–Stokes equations. Moreover, we do so by showing a sequence of difference quotients converges to the unique solution of the sensitivity equations for both the 2D Navier–Stokes equations and the related data assimilation equations, which utilize the continuous data assimilation algorithm proposed by Azouani, Olson, and Titi. As a result, this method of proof provides uniform bounds on difference quotients, demonstrating parameter recovery algorithms that change parameters as the system evolves will be well behaved. Furthermore, our analysis can be extended to analyze the sensitivity of the 2D Euler equations to a viscous regularization. We also note that this appears to be the first such rigorous proof of global existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier–Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the viscosity.
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Notes
Note that considerations of sensitivity arise in the context of perturbations; hence, the natural initial data for a sensitivity equation is identically zero data.
References
Albanez, D.A., Nussenzveig Lopes, H.J., Titi, E.S.: Continuous data assimilation for the three-dimensional Navier–Stokes- \(\alpha \) model. Asymptotic Anal. 97(1–2), 139–164 (2016)
Altaf, M.U., Titi, E.S., Knio, O.M., Zhao, L., McCabe, M.F., Hoteit, I.: Downscaling the 2D Benard convection equations using continuous data assimilation. Comput. Geosci 21(3), 393–410 (2017)
Anderson, K., Newman, J.C., Whitfield, D.L., Nielsen, E.J.: Sensitivity analysis for Navier–Stokes equations on unstructured meshes using complex variables. AIAA J. 39, 11 (1999)
Azouani, A., Titi, E.S.: Feedback control of nonlinear dissipative systems by finite determining parameters-a reaction-diffusion paradigm. Evol. Equ. Control Theory 3(4), 579–594 (2014)
Azouani, A., Olson, E., Titi, E.S.: Continuous data assimilation using general interpolant observables. J. Nonlinear Sci. 24(2), 277–304 (2014)
Bessaih, H., Olson, E., Titi, E.S.: Continuous data assimilation with stochastically noisy data. Nonlinearity 28(3), 729–753 (2015)
Biswas, A., Martinez, V.R.: Higher-order synchronization for a data assimilation algorithm for the 2D Navier–Stokes equations. Nonlinear Anal. Real World Appl. 35, 132–157 (2017)
Biswas, A., Hudson, J., Larios, A., Pei, Y.: Continuous data assimilation for the 2D magnetohydrodynamic equations using one component of the velocity and magnetic fields. Asymptot. Anal. 108(1–2), 1–43 (2018)
Biswas, A., Foias, C., Mondaini, C.F., Titi, E.S.: Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations. In: Annales de l’Institut Henri Poincaré C, Analyse non linéaire, pp. 295–326. Elsevier (2019)
Blömker, D., Law, K., Stuart, A.M., Zygalakis, K.C.: Accuracy and stability of the continuous-time 3DVAR filter for the Navier–Stokes equation. Nonlinearity 26(8), 2193–2219 (2013)
Borggaard, J., Burns, J.: A PDE sensitivity equation method for optimal aerodynamic design. J. Comput. Phys. 136(2), 366–384 (1997)
Breckling, S., Neda, M., Pahlevani, F.: A sensitivity study of the Navier–Stokes- \(\alpha \) model. Comput. Math. Appl. 75(2), 666–689 (2018)
Brewer, D.: The differentiability with respect to a parameter of the solution of a linear abstract Cauchy problem. SIAM J. Math. Anal. 13(4), 607–620 (1982)
Carlson, E., Larios, A.: Super-exponential convergence of certain nonlinear algorithms for continuous data assimilation. (2020). (preprint)
Carlson, E., Hudson, J., Larios, A.: Parameter recovery for the 2 dimensional Navier–Stokes equations via continuous data assimilation. SIAM J. Sci. Comput. 42(1), A250–A270 (2020)
Celik, E., Olson, E., Titi, E.S.: Spectral filtering of interpolant observables for a discrete-in-time downscaling data assimilation algorithm. SIAM J. Appl. Dyn. Syst. 18(2), 1118–1142 (2019)
Constantin, P., Foias, C.: Global Lyapunov exponents, Kaplan–Yorke formulas and the dimension of the attractors for 2D Navier–Stokes equations. Commun. Pure Appl. Math. 38(1), 1–27 (1985)
Constantin, P., Foias, C.: Navier–Stokes Equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL (1988)
Davis, L., Pahlevani, F.: Parameter sensitivity of an eddy viscosity model: analysis, computation and its application to quantifying model reliability. Int. J. Uncertain. Quantif. 3(5), 397–419 (2013)
Desamsetti, S., Dasari, H.P., Langodan, S., Knio, O., Hoteit, I., Titi, E.S.: Efficient dynamical downscaling of general circulation models using continuous data assimilation. Q. J. R. Meteorol. Soc. 145, 3175–3194 (2019). https://doi.org/10.1002/qj.3612. http://hdl.handle.net/10754/656325
Di Leoni, P.C., Mazzino, A., Biferale, L.: Inferring flow parameters and turbulent configuration with physics-informed data assimilation and spectral nudging. Phys. Rev. Fluids 3(10), 104604 (2018)
Farhat, A., Jolly, M.S., Titi, E.S.: Continuous data assimilation for the 2D Bénard convection through velocity measurements alone. Phys. D 303, 59–66 (2015)
Farhat, A., Lunasin, E., Titi, E.S.: Abridged continuous data assimilation for the 2D Navier–Stokes equations utilizing measurements of only one component of the velocity field. J. Math. Fluid Mech. 18(1), 1–23 (2016)
Farhat, A., Lunasin, E., Titi, E.S.: Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements. J. Math. Anal. Appl. 438(1), 492–506 (2016)
Farhat, A., Lunasin, E., Titi, E.S.: On the Charney conjecture of data assimilation employing temperature measurements alone: the paradigm of 3D planetary geostrophic model. Math. Clim. Weather Forecast. 2(1), 61–74 (2016). https://doi.org/10.1515/mcwf-2016-0004
Farhat, A., Lunasin, E., Titi, E.S.: Continuous data assimilation for a 2D Bénard convection system through horizontal velocity measurements alone. J. Nonlinear Sci., 1–23 (2017)
Farhat, A., Johnston, H., Jolly, M., Titi, E.S.: Assimilation of nearly turbulent Rayleigh–Bénard flow through vorticity or local circulation measurements: a computational study. J. Sci. Comput. 77(3), 1519–1533 (2018)
Farhat, A., Lunasin, E., Titi, E.S.: A data assimilation algorithm: the paradigm of the 3D Leray-\(\alpha \) model of turbulence. In: Partial Differential Equations Arising from Physics and Geometry, vol. 450, pp. 253–273 (2019)
Farhat, A., Glatt-Holtz, N.E., Martinez, V.R., McQuarrie, S.A., Whitehead, J.P.: Data assimilation in large Prandtl Rayleigh–Bénard convection from thermal measurements. SIAM J. Appl. Dyn. Syst. 19(1), 510–540 (2020)
Fernández, M.A., Moubachir, M.: Sensitivity analysis for an incompressible aeroelastic system. Math. Models Methods Appl. Sci. 12, 1109–1130 (2002)
Foias, C., Manley, O., Rosa, R., Temam, R.: Navier–Stokes Equations and Turbulence. Encyclopedia of Mathematics and its Applications, vol. 83. Cambridge University Press, Cambridge (2001)
Foias, C., Mondaini, C.F., Titi, E.S.: A discrete data assimilation scheme for the solutions of the two-dimensional Navier–Stokes equations and their statistics. SIAM J. Appl. Dyn. Syst. 15(4), 2109–2142 (2016)
Foyash, K., Dzholli, M.S., Kravchenko, R., Titi, È.S.: A unified approach to the construction of defining forms for a two-dimensional system of Navier–Stokes equations: the case of general interpolating operators. Uspekhi Mat. Nauk 69(2(416)), 177–200 (2014)
García-Archilla, B., Novo, J., Titi, E.S.: Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the Navier-Stokes equations. SIAM J. Numer. Anal. 58(1), 410–429 (2020)
Gardner, M., Larios, A., Rebholz, L.G., Vargun, D., Zerfas, C.: Continuous data assimilation applied to a velocity-vorticity formulation of the 2d Navier–Stokes equations. Electron. Res. Arch. 29(3), 2223–2247 (2021)
Gesho, M., Olson, E., Titi, E.S.: A computational study of a data assimilation algorithm for the two-dimensional Navier–Stokes equations. Commun. Comput. Phys. 19(4), 1094–1110 (2016)
Gibson, J., Clark, L.: Sensitivity analysis for a class of evolution equations. J. Comput. Phys. 136(2), 366–384 (1997)
Glatt-Holtz, N., Kukavica, I., Vicol, V., Ziane, M.: Existence and regularity of invariant measures for the three dimensional stochastic primitive equations. J. Math. Phys. 55(5), 051504, 34 (2014)
Golovkin, K.K.: Vanishing viscosity in Cauchy’s problem for hydrodynamics equations. Proc. Steklov Inst. Math. 92, 33–53 (1966)
Grappin, R., Léorat, J.: Lyapunov exponents and the dimension of periodic incompressible Navier–Stokes flows: numerical measurements. J. Fluid Mech. 222, 61–94 (1991)
Hamby, D.: A review of techniques for parameter sensitivity analysis of environmental models. Environ. Monit. Assess. 32(2), 135–154 (1994)
Hudson, J., Jolly, M.: Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations. J. Comput. Dyn. 6(1), 131–145 (2019)
Ibdah, H.A., Mondaini, C.F., Titi, E.S.: Fully discrete numerical schemes of a data assimilation algorithm: uniform-in-time error estimates. IMA J. Numer. Anal. 11 2019. drz043
Jolly, M.S., Martinez, V.R., Titi, E.S.: A data assimilation algorithm for the subcritical surface quasi-geostrophic equation. Adv. Nonlinear Stud. 17(1), 167–192 (2017)
Jolly, M.S., Martinez, V.R., Olson, E.J., Titi, E.S.: Continuous data assimilation with blurred-in-time measurements of the surface quasi-geostrophic equation. Chin. Ann. Math. Ser. B 40(5), 721–764 (2019)
Kato, T.: Nonstationary flows of viscous and ideal fluids in \({ R}^{3}\). J. Funct. Anal. 9, 296–305 (1972)
Kim, H., Kim, C., Rho, O.-H., Dong Lee, K.: Aerodynamic sensitivity analysis for Navier–Stokes equations. J. KSIAM 3, 161–171 (1999)
Kouhi, M., Houzeaux, G., Cucchietti, F., Vázquez, M.: Implementation of discrete adjoint method for parameter sensitivity analysis in chemically reacting flows. In: 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA SciTech Forum (2016)
Larios, A., Pei, Y.: Nonlinear continuous data assimilation. (2017). arXiv:1703.03546
Larios, A., Victor, C.: Continuous data assimilation with a moving cluster of data points for a reaction diffusion equation: a computational study. Commun. Comput. Phys. 29, 1273–1298 (2021)
Larios, A., Rebholz, L.G., Zerfas, C.: Global in time stability and accuracy of IMEX-FEM data assimilation schemes for Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 345, 1077–1093 (2018). https://doi.org/10.1016/j.cma.2018.09.004. http://www.sciencedirect.com/science/article/pii/S004578251830447X
Leoni, D., Clark, P., Mazzino, A., Biferale, L.: Synchronization to big-data: nudging the Navier–Stokes equations for data assimilation of turbulent flows. arXiv preprint arXiv:1905.05860, (2019)
Lunasin, E., Titi, E.S.: Finite determining parameters feedback control for distributed nonlinear dissipative systems-a computational study. Evol. Equ. Control Theory 6(4), 535–557 (2017)
Markowich, P.A., Titi, E.S., Trabelsi, S.: Continuous data assimilation for the three-dimensional Brinkman–Forchheimer-extended Darcy model. Nonlinearity 29(4), 1292–1328 (2016)
Masmoudi, N.: Remarks about the inviscid limit of the Navier–Stokes system. Commun. Math. Phys. 270, 777–788 (2007)
Mondaini, C.F., Titi, E.S.: Uniform-in-time error estimates for the postprocessing Galerkin method applied to a data assimilation algorithm. SIAM J. Numer. Anal. 56(1), 78–110 (2018)
Neda, M., Pahlevani, F., Rebholz, L.G., Waters, J.: Sensitivity analysis of the grad-div stabilization parameter in finite element simulations of incompressible flow. J. Numer. Math. 24(3), 189–206 (2016)
Noacco, V., Sarrazin, F., Pianosi, F., Wagener, T.: Matlab/r workflows to assess critical choices in global sensitivity analysis using the safe toolbox. MethodsX 6, 2258–2280 (2019)
Pahlevani, F.: Sensitivity analysis of eddy viscosity models. ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–University of Pittsburgh
Pahlevani, F.: Sensitivity computations of eddy viscosity models with an application in drag computation. Int. J. Numer. Methods Fluids 52(4), 381–392 (2006)
Pei, Y.: Continuous data assimilation for the 3D primitive equations of the ocean. Commun. Pure Appl. Math. 18(2), 643 (2019)
Ponce, G.: On two-dimensional incompressible fluids. Comm. Part. Diff. Eqs. 11, 483–511 (1986)
Rebholz, L.G., Zerfas, C.: Simple and efficient continuous data assimilation of evolution equations via algebraic nudging. Numer. Methods Partial Differ. Equ. 37, 1–25 (2021)
Rebholz, L., Zerfas, C., Zhao, K.: Global in time analysis and sensitivity analysis for the reduced NS-\(\alpha \) model of incompressible flow. J. Math. Fluid Mech. 19(3), 445–467 (2017)
Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors
Singler, J.: Differentiability with respect to parameters of weak solutions of linear parabolic equations. Math. Comput. Model. 47(3), 422–430 (2008)
Stanley, L.G., Stewart, D.L.: Design Sensitivity Analysis: Computational Issues of Sensitivity Equation Methods, volume 25 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Computational issues of sensitivity equation methods
Swann, H.G.: The convergence with vanishing viscosity of nonstationary Navier–Stokes flow to ideal flow in \({\mathbb{R}}^3\). Trans. Am. Math. Soc. 157, 373–397 (1971)
Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis, volume 66 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition (1995)
Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing, Providence, RI, 2001. Theory and numerical analysis, Reprint of the 1984 edition
Vemuri, V., Raefsky, A.: On a new approach to parameter estimation by the method of sensitivity functions. Int. J. Syst. Sci. 10(4), 395–407 (1979)
Zerfas, C., Rebholz, L.G., Schneier, M., Iliescu, T.: Continuous data assimilation reduced order models of fluid flow. Comput. Methods Appl. Mech. Eng. 357, 112596, 18 (2019)
Acknowledgements
The authors would like to thank the anonymous referees for helpful comments and suggestions that significantly improved the quality of this manuscript. E.C. would like to give thanks for the kind hospitality of the COSIM group at Los Alamos National Laboratory where some of this work was completed. The research of E.C. was supported in part by the NSF GRFP grant no. 1610400. The research of A.L. was supported in part by the NSF grants no. DMS-1716801 and CMMI-1953346.
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Communicated by Alain Goriely.
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Carlson, E., Larios, A. Sensitivity Analysis for the 2D Navier–Stokes Equations with Applications to Continuous Data Assimilation. J Nonlinear Sci 31, 84 (2021). https://doi.org/10.1007/s00332-021-09739-9
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DOI: https://doi.org/10.1007/s00332-021-09739-9